Drop Rate Geometric Attempts To Success

Compute independent constant-probability first-success expectation and finite confidence quantile, with stable tails and no promise that the expectation is a guarantee; no pity/dependence assumed.

Description

A geometric waiting model assumes an independent fixed success probability on every attempt.

A geometric waiting model assumes an independent fixed success probability on every attempt. A confidence target asks for a minimum whole draw count.^[1]

Enter p strictly above 0 through 1 and confidence 0–1. Confidence 0 needs zero draws; p 1 needs one for positive confidence. Confidence 1 with p<1 is an input error. Use log 1 p/expm 1 for stable results; return safe-integer draws, mean wait and achieved confidence.

When to use Drop Rate Geometric Attempts To Success

Use Drop Rate Geometric Attempts To Success when you need to evaluate the supplied records under the calculation and evidence rules below. No pity, changing probabilities or guaranteed finite success when p<1 is modeled.

Prepare the supplied game state, observations and explicit variant settings. Follow the card, board, coordinate and state conventions in the calculation rules.

How to use Drop Rate Geometric Attempts To Success and understand its results

Prepare the supplied game state, observations and explicit variant settings. Follow the card, board, coordinate and state conventions in the calculation rules.
Enter independentSuccessProbability, callerTargetSuccessConfidence using the field table and calculation rules, then run the tool.
Read expectedAttemptsToFirstSuccess, minimumAttemptsAtCallerConfidence, attainedAtLeastOneSuccessProbability. The explanation below defines the units, signs, ordering and any flags or placeholder values.
If the tool reports an input issue, correct the values and run it again. Apply the stated assumptions when interpreting a successfully returned result.

Input field	Meaning	Value type; see rules below	Requirement
independentSuccessProbability	Independent success probability	number	Required
callerTargetSuccessConfidence	Caller target success confidence	number	Required

P(success by n)=1−(1−p)^n; n=ceil(log(1−confidence)/log(1−p))

Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.

Output field	Meaning	Value type
expectedAttemptsToFirstSuccess	Expected attempts to first success	number
minimumAttemptsAtCallerConfidence	Minimum attempts at caller confidence	number
attainedAtLeastOneSuccessProbability	Attained at least one success probability	number

json

{
  "input": {
    "independentSuccessProbability": 0.5,
    "callerTargetSuccessConfidence": 0.75
  },
  "output": {
    "expectedAttemptsToFirstSuccess": 2,
    "minimumAttemptsAtCallerConfidence": 2,
    "attainedAtLeastOneSuccessProbability": 0.75
  }
}

Limitations and assumptions

No pity, changing probabilities or guaranteed finite success when p<1 is modeled.

Numeric inputs, where present, must be finite. Follow the calculation-specific rules for overflow, underflow and exceptional cases. Missing required fields, extra fields, wrong value types and violations of the calculation rules are input errors.

Alternative or Complementary methods

Use Gacha Pity Counter Expected Cost: A conditional pity schedule gives success probability after each failure count. Certainty at the final step bounds the remaining wait. For a complementary task, use Loot Box Expected Cost Per Item: Collecting all missing items takes longer when draws repeat owned items. A subset expectation accounts for unequal item probabilities.

References

Drop Rate Geometric Attempts To Success — tool.io

Similar or alternative tools

Gacha Pity Counter Expected Cost
A conditional pity schedule gives success probability after each failure count. Certainty at the final step bounds the remaining wait.
Loot Box Expected Cost Per Item
Collecting all missing items takes longer when draws repeat owned items. A subset expectation accounts for unequal item probabilities.